Abstract
Let G be a semisimple Lie group of rank ⩾2 and Γ an irreducible lattice. Γ has two natural metrics: a metric inherited from a Riemannian metric on the ambient Lie group and a word metric defined with respect to some finite set of generators. Confirming a conjecture of D. Kazhdan (cf. Gromov [Gr2]) we show that these metrics are Lipschitz equivalent. It is shown that a cyclic subgroup of Γ is virtually unipotent if and only if it has exponential growth with respect to the generators of Γ.
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Lubotzky, A., Mozes, S. & Raghunathan, M.S. The word and Riemannian metrics on lattices of semisimple groups. Publications Mathématiques de l’Institut des Hautes Scientifiques 91, 5–53 (2000). https://doi.org/10.1007/BF02698740
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DOI: https://doi.org/10.1007/BF02698740